Solving some q-trigonometric conjectures of Gosper

Gosper in 2001 introduced the functions sinq⁡z and cosq⁡z as q-analogues for the trigonometric functions sin⁡z and cos⁡z respectively. He stated, but did not prove, many identities involving the functions sinq⁡z and cosq⁡z along with some other related functions and constants. In this note we shall provide proofs for some of these identities by means of the classical theory of elliptic functions.